Elliptic curves from finite order recursions or non-involutive permutations for discrete dynamical systems and lattice statistical mechanics

Abstract

We study birational mappings generated by matrix inversion and permutations of the entries of matrices. For q=3 we have performed a systematic examination of all the birational mappings associated with permutations of matrices in order to find integrable mappings and some finite order recursions. This exhaustive analysis gives, among 30 462 classes of mappings, 20 classes of integrable birational mappings, 8 classes associated with integrable recursions and 44 classes yielding finite order recursions. An exhaustive analysis (with a constraint on the diagonal entries) has also been performed for matrices: we have found 880 new classes of mappings associated with integrable recursions. We have visualized the orbits of the birational mappings corresponding to these 880 classes. Most correspond to elliptic curves and very few to surfaces or higher dimensional algebraic varieties. All these new examples show that integrability can actually correspond to non-involutive permutations. The analysis of the integrable cases specific of a particular size of the matrix and a careful examination of the non-involutive permutations, shed some light on the integrability of such birational mappings.